This note introduces a new general conjecture correlating the dimensionality dT of an infinite lattice with N nodes to the asymptotic value of its Wiener Index W(N). In the limit of large N the general asymptotic behavior W(N)≈Ns is proposed, where the exponent s and dT are related by the conjectured formula s=2+1/dT allowing a new definition of dimensionality dW=(s-2)-1. Being related to the t...

We prove that if (H,G) is a small, nm-stable compact G-group, then H is nilpotent-by-finite, and if additionally NM(H) ≤ ω, then H is abelian-byfinite. Both results are significant steps towards the proof of the conjecture that each small, nm-stable compact G-group is abelian-by-finite. We provide counter-examples to the NM-gap conjecture, that is we give examples of small, nm-stable compact G-...

We consider 4d and 5d N = 2 supersymmetric theories and demonstrate that in general their SeibergWitten prepotentials satisfy the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations. General proof for the Yang-Mills models (with matter in the first fundamental representation) makes use of the hyperelliptic curves and underlying integrable systems. A wide class of examples is discussed, it conta...

The fractional arboricity of a digraph D $D$ , denoted by γ ( ) $\gamma (D)$ is defined as = max H ⊆ | V > 1 A − (D)=\mathop{\max }\limits_{H\subseteq D,|V(H)|\gt 1}\frac{|A(H)|}{|V(H)|-1}$ . Frank proved that decomposes into k $k$ branchings, if and only Δ ≤ ${{\rm{\Delta }}}^{-}(D)\le k$ (D)\le In this paper, we study analogues for the Nine Dragon Tree Conjecture. We conjecture that, positive...

The Fourier Entropy-Influence (FEI) conjecture of Friedgut and Kalai [1] seeks to relate two fundamental measures of Boolean function complexity: it states that H[f ] ≤ C · Inf [f ] holds for every Boolean function f , where H[f ] denotes the spectral entropy of f , Inf [f ] is its total influence, and C > 0 is a universal constant. Despite significant interest in the conjecture it has only bee...

A simple graph H is a cover of a graph G if there exists a mapping φ from H onto G such that φ maps the neighbors of every vertex v in H bijectively to the neighbors of φ(v) in G. Negami conjectured in 1986 that a connected graph has a finite planar cover if and only if it embeds in the projective plane. The conjecture is still open. It follows from the results of Archdeacon, Fellows, Negami, a...

A cubic graph H is called a Kotzig graph if H has a circuit double cover consisting of three Hamilton circuits. It was first proved by Goddyn that if a cubic graph G contains a spanning subgraph H which is a subdivision of a Kotzig graph then G has a circuit double cover. A spanning subgraph H of a cubic graph G is called a Kotzig frame if the contracted graph G/H is even and every non-circuit ...

The two most fundamental conjectures on the structure of generic Hecke algebra $\mathcal{H}(W)$ associated with a complex reflection group $W$ state that is free module rank $|W|$ over its ring definition, and admits canonical symmetrising trace. first conjecture has recently become theorem, while second conjecture, known to hold for real groups, only been proved some exceptional non-real group...

We present a conjecture which unifies several conjectures on motives in characteristic p. Introduction In a talk at the June 1996 Oberwolfach algebraic K-theory conference, I explained that, in view of Voevodsky’s proof of the Milnor conjecture, the Bass conjecture on finite generation of K-groups of regular schemes of finite type over the integers implies the rational Beilinson-Soulé conjectur...